Double flag varieties for a symmetric pair and finiteness of orbits

Kyo Nishiyama, Hiroyuki Ochiai

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Let G be a reductive algebraic group over the complex number filed, and K = Gθ be the fixed points of an involutive automorphism θ of G so that (G,K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal Kaction. The double flag variety G/P × K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P1 × G/P2 × G/P3 is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. In this paper, we study double flag varieties G/P × K/Q of finite type. We give efficient criterion under which the double flag variety is of finite type. The finiteness of orbits is strongly related to spherical actions of G or K. For example, we show a partial flag variety G/P is K-spherical if a product of partial flag varieties G/P ×G/θ(P) is G-spherical. We also give many examples of the double flag varieties of finite type, and for type AIII, we give a classification when P = B is a Borel subgroup of G.

Original languageEnglish
Pages (from-to)79-99
Number of pages21
JournalJournal of Lie Theory
Volume21
Issue number1
Publication statusPublished - 2011

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Flag Variety
Finiteness
Orbit
Finite Type
Partial
Parabolic Subgroup
Reductive Group
Algebraic Groups
Complex number
Automorphism
Fixed point
Subgroup

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Double flag varieties for a symmetric pair and finiteness of orbits. / Nishiyama, Kyo; Ochiai, Hiroyuki.

In: Journal of Lie Theory, Vol. 21, No. 1, 2011, p. 79-99.

Research output: Contribution to journalArticle

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